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I will take a stab at this. So we have a circle at each point of Minkowski space, S^1. Say at the origin we have magnetic charge. Suppose we now make one full orbit of this charge in the plane z = 0. Suppose that when we come back to where we started in the z = 0 plane in the compact space we have advanced or retarded by one full turn in the space S^1? Is that what topological twisting of S^1 means?

The magnetic charge is the topological twisting?

There is something else we could do. Suppose we take the circular fibers, give them a cut and then glue the ends at different points of our base space, let each pair of ends of all the circles be separated by the same small spacetime vector? Once around the fiber and we do not come back to where we started in space or even time. Is that an operation that gives a topological twisting of S^1?

You want the resulting space (4D Minkowski) to behave as if it were cosmological in nature i.e. you cannot choose a unique point to represent the origin.
It would be like saying that the earths position is special compared to the rest of the universe.

If we allow such an origin it would be hard to talk about "our physics" unless I'm attributing too much importance to isotropy/homogeneity.

I was not clear then. I am suggesting that we look at the space surrounding a point magnetic charge and for simplicity let us place it at the origin of some coordinate system, just as we could place a point charge at the origin. There are only so many ways one can topologically twist the fibers S^1?

( And there has to be a whole lot of symmetry in the twisting of the fibers S^1, the symmetry of a point magnetic charge?)